For a given positive integer m m , let A = { 0 , 1 , … , m } A=\{0,1,\ldots ,m\} and q ∈ ( m , m + 1 ) q \in (m,m+1) . A sequence ( c i ) = c 1 c 2 … (c_i)=c_1c_2 \ldots consisting of elements in A A is called an expansion of x x if ∑ i = 1 ∞ c i q − i = x \sum _{i=1}^{\infty } c_i q^{-i}=x . It is known that almost every x x belonging to the interval [ 0 , m / ( q − 1 ) ] [0,m/(q-1)] has uncountably many expansions. In this paper we study the existence of expansions ( d i ) (d_i) of x x satisfying the inequalities ∑ i = 1 n d i q − i ≥ ∑ i = 1 n c i q − i \sum _{i=1}^n d_iq^{-i} \ge \sum _{i=1}^n c_i q^{-i} , n = 1 , 2 , … , n=1,2,\ldots , for each expansion ( c i ) (c_i) of x x .